Annals o Ope a ions Resea ch
h ps://doi.o g/10.1007/s10479-023-05601-1
ORIGINAL RESEARCH
Co po a e social esponsibili y and ne wo k ex e nali ies:
a game- heo e ic app oach
Domenico Buccella1·Luciano Fan i2·Luca Go i3·Mau o Sodini4,5
Recei ed: 26 Janua y 2023 / Accep ed: 8 Sep embe 2023
© The Au ho (s) 2023
Abs ac
This esea ch e isi s he pionee ing wo k by Ka z and Shapi o (Am Econom Re
75:424–440, 1985) wi h ne wo k (consump ion) ex e nali ies in a wo old way: i s , i con-
side s Co po a e Socially Responsible (CSR), ins ead o p o i -maximising, i ms; second,
i uses a game- heo e ic app oach and analyses he commi men decision game in which
i ms ace he bina y choice o c edibly commi (C) o no o commi (NC) hemsel es oan
announced ou pu le el in he i s decision-making s age. Compe i ion a he ma ke s age
occu s à la Cou no . Resul s show a ich spec um o sub-game pe ec Nash equilib ium
(SPNE) ou comes, anging om he p isone ’s dilemma (sel -in e es and mu ual bene i o
ou pu commi men con lic ) o he an i-p isone ’s dilemma o deadlock (sel -in e es and
mu ual bene i o ou pu commi men do no con lic ), passing om he coo dina ion o he
an i-coo dina ion game. These ou comes depend on he in ensi y o he social conce n in
he i m’s objec i e and he ne wo k size. The a icle also pinpoin s he wel a e ou comes
co esponding o he SPNE and ex ends he analysis o a S ackelbe g i al y se ing.
BLuca Go i
[email p o ec ed]; d [email p o ec ed]
Domenico Buccella
[email protected]
Luciano Fan i
[email p o ec ed]
Mau o Sodini
[email p o ec ed]
1Depa men o Economics, Kozminski Uni e si y, Jagiello´nska S ee , 57/59, 03301 Wa saw,
Poland
2Depa men o Economics and Managemen , Uni e si y o Pisa, Via Cosimo Ridol i, 10,
I–56124 Pisa (PI), I aly
3Depa men o Law, Uni e si y o Pisa, Via Collegio Ricci, 10, I–56126 Pisa (PI), I aly
4Depa men o Law, Uni e si y o Naples “Fede ico II”, Via Mezzocannone 16, I–80134 Naples
(NA), I aly
5Depa men o Finance, Facul y o Economics, Technical Uni e si y o Os a a, Os a a, Czech
Republic
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Annals o Ope a ions Resea ch
Keywo ds Ne wo k ex e nali y ·Cou no and S ackelbe g duopolies ·Commi men ·
Co po a e social esponsibili y
JEL Classi ica ion H23 ·L13 ·O31
1 In oduc ion
This a icle examines a quan i y-se ing (Cou no ) duopoly wi h Co po a e Socially Respon-
sible (CSR) i ms and ne wo k ex e nali ies in consump ion,1 ep esen ing wo pilla s o
mode n indus ies amed in a s a egic con ex . I ollows he pionee ing wo k by Ka z &
Shapi o (1985), who exogenously dis inguish he cases in which i ms commi and do no
commi o a publicly announced ou pu le el, and s udies he commi men decision game
wi h CSR ins ead o p o i -maximising i ms, which has ecen ly been conside ed by Choi &
Lim (2022). The a icle belongs o a li e a u e on he opic o he iming o commi men and
expec a ions o ma ion mechanisms in s a egic ma ke s wi h ne wo k e ec s (e.g., G i a &
Ve as, 2011; Hagiu & Halabu da, 2014; Leppänen, 2020; Nakamu a, 2021; Suleymano a &
Wey, 2012; Toshimi su, 2019,2021).
Engagemen in CSR has ecen ly eme ged as a dominan global business p ac ice and has
been he esul o wo main o ces. On one hand, he expec a ions o he public audience and
he o e all socie y o business a e ising. As McKinsey (2019) epo s, cus ome in e es in
being in o med abou companies’ engagemen in en i onmen al and social issues has g own;
cus ome s also s ongly belie e ha companies ha e a esponsibili y on hose subjec s. On he
o he hand,aglobalsu eyo 350businessleade sconduc edbyDeloi e(2019)(incollabo a-
ionwi hFo bes) e eals ha businessleade sa edeeplycon incedo oleo i msass ewa ds
o socie y, and hey a e planning o engage mo e on socie al-impac issues in he nex u u e.
CSR has also become an issue o bu geoning impo ance in he academic li e a u e
(e.g., Ba on, 2001,2009; Goe ing, 2007,2008; Ga cía-Gallego & Geo gan zís, 2009;
Lambe ini & Tampie i, 2010,2015; Bénabou & Ti ole, 2010;Kopele al.,2014; Kopel
& Laman ia, 2018). Amongs he indus ies ha ha e expe ienced ex ao dina y p og ess
in CSR ac i i ies in ecen yea s, i is ema kable o ind a leading posi ion p ecisely in
hose p oducing ne wo k goods. Fo ins ance, acco ding o a KPMG epo , he echnology,
media & elecommunica ions sec o p esen s 79 pe cen o he companies su eyed
epo ing CSR ac i i ies, he highes le el amongs he su eyed indus ies. Speci ically, he
elecommunica ion subsec o shows he highes a e o CSR epo ing, wi h 87 pe cen o
he companies (KPMG, 2015,2017).
The ema kable de elopmen pace o ne wo k indus ies has led an inc easing numbe o
schola s o s a s udying how consump ion ex e nali ies may al e he esul s o he s an-
da d models o impe ec compe i ion amed in s a egic con ex s (Ka z & Shapi o, 1985),
ocusing also on he ole o monopoly (Cab al e al., 1990), R&D inno a ion (Buccella e al.,
2022a;Cab al,1990;Naska &Pal,2020;Sh i as a ,2021) s a egic delega ion (Bha acha -
jee & Pal, 2014; Chi co & Sc imi o e, 2013;Hoe nig,2012), and unionised labou ma ke
and code e mina ion (Fan i & Buccella, 2016; Fan i & Go i, 2019).
The beha iou o i ms unde ne wo k consump ion ex e nali ies is wo old (Ka z &
Shapi o, 1985): hey can o canno commi o a publicly known ou pu le el. This choice, in
1Ne wo k goods can gene a e posi i e ( esp. nega i e) ex e nali ies, in u n, causing he so-called bandwagon
( esp. snob) e ec : he desi e o ha e ( esp. no o ha e) some goods because almos e e yone else has hem.
A ypical example is he posi i e ( esp. nega i e) ea u e o ashion due o widesp ead b ands.
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Annals o Ope a ions Resea ch
u n, a ec s he size o he ma ke , he consume wel a e, p o i s and social wel a e. I i ms
canno c edibly commi hemsel es o an announced ou pu le el, each o hem ollows he
s anda d Cou no assump ion ha he ac ual ou pu o he i al is ixed and hen i chooses i s
p oduc ion by assuming ha he consume s’ expec a ions abou he size o he ne wo k a e
gi en. In his case “a i m’s announcemen o i s planned le el o ou pu does no a ec con-
sume expec a ions” (Ka z & Shapi o, 1985, p. 440). I i ms can c edibly commi hemsel es
o an announced ou pu le el, consume s know he selec ed ou pu and hen expec a ions a e
equal o he commi ed p oduc ion. The key di e ence be ween hese cases is ha unde ull
commi men “ i m ican di ec ly in luence consume s’ expec a ions ega ding i s ne wo k
size”, hus ge ing “ he s anda d Cou no equilib ium wi h demand-side economies o scale”
(Ka z & Shapi o, 1985, p. 440).
The p esen a icle ex ends Ka z & Shapi o (1985) in a wo old way: (1) i conside s CSR,
ins ead o p o i -maximising, i ms, and (2) i uses a game- heo e ic app oach o analyse
he CSR commi men decision game (CSR-CDG hence o h) in which i ms ace he bina y
choice o commi (C)o no ocommi (NC) hemsel es o an announced ou pu le el in
he i s (decision-making) s age, and hen de e mine he endogenous ma ke con igu a ion.
These wo scena ios (Cand NC) we e aken exogenously and analysed sepa a ely in he
o iginal con ibu ion o Ka z & Shapi o (1985). Howe e , his choice has impo an s a egic
e ec s, wi h implica ions on he endogenous eme gence o sub-game pe ec Nash equi-
lib ium (SPNE) and social wel a e e ec s, which ins ead de e mine some possible policy
implica ions. These e ec s ha e been poin ed ou only ecen ly by Choi & Lim (2022)in
a ele an con ibu ion s udying he CDG wi h p o i -maximising i ms. Dealing wi h CSR
i ms, he p esen con ibu ion shows sha p di e ences in he p e ailing SPNE compa ed he
esul s ound by Choi & Lim (2022). This is because o he s a egic e ec s gene a ed by he
ex en o he deg ee o social conce n in he i m’s objec i e.
I i ms a e p o i -maximising (Choi & Lim, 2022), he ou comes o he CDG a e he
ollowing:
•I he ne wo k ex e nali y is posi i e, he unique Pa e o ine icien SPNE is (C,C).The
CDG is a p isone ’s dilemma in which sel -in e es and mu ual bene i o ou pu commi -
men con lic when he ex en o he nega i e ne wo k e ec is su icien ly low (n<0.75),
whe e −1<n<1 is he s eng h o he consump ion ne wo k ex e nali y. The CDG
becomes an an i-p isone ’s dilemma (deadlock) in which sel -in e es and mu ual bene i
o ou pu commi men do no con lic when he ex en o he nega i e ne wo k e ec is
su icien ly high (n>0.75). Unde ou pu commi men i ms p oduce mo e han unde
ou pu non-commi men due o he bandwagon e ec gene a ed by he posi i e ne wo k
ex e nali y. In his case, he commi men s a egy ends o incen i ise i ms o p oduce a
highe ou pu han he non-commi men s a egy because o he la ge (ou wa d) shi in he
ma ke demand o he ne wo k goods gene a ed by he posi i e ex e nali y ha commi ing
in ad ance, in u n, causes compa ed o non-commi ing in ad ance o a gi en ou pu le el.
I he ne wo k e ec is ela i ely low, Cis a dominan s a egy bu i ms a e en apped in a
dilemma as hey could join ly ob ain a highe p o i by playing NC, bu nei he would like
o be he one ha unila e ally ob ains he lowes payo h ough he NC s a egy. La ge
alues o he ne wo k e ec allow i ms o sol e he dilemma as he inc ease in he ou pu
sus ained by he highe demand inc eases p o i s unde he Cs a egy le ing hem when
bo h i ms play Cgo beyond he co esponding alue i bo h played NC. Commi ing o
an announced ou pu le el, he e o e, has a ele an s a egic alue in a non-coope a i e
con ex as, e.g., he manage ial delega ion de ice.
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Annals o Ope a ions Resea ch
•I he ne wo k ex e nali y is nega i e, he unique Pa e o ine icien SPNE is (NC,NC)
and he CDG is a p isone ’s dilemma in which sel -in e es and mu ual bene i o ou pu
non-commi men con lic . This holds i espec i e o he ex en o he nega i e ne wo k
e ec . Unde ou pu commi men i ms p oduce less han unde ou pu non-commi men
due o he snob e ec gene a ed by he nega i e ne wo k ex e nali y. In his case, he
non-commi men s a egy ends o incen i ise i ms o p oduce a highe ou pu han he
commi men s a egy because o he la ge (inwa d) shi in he ma ke demand o he
ne wo k goods gene a ed by he nega i e ex e nali y ha commi ing in ad ance, in u n,
causes compa ed o non-commi ing in ad ance o a gi en ou pu le el. Gi en he snob
e ec , NC is a dominan s a egy bu i ms a e en apped in a dilemma as hey could join ly
ob ain a highe p o i by playing C, bu nei he would like o be he one ha unila e ally
ob ains he lowes payo h ough he Cs a egy.
I i ms a e socially o ien ed (CSR), he e a e ele an changes in he p e ailing SPNE
o he CSR-CDG compa ed o Choi & Lim (2022). An inc ease in he weigh o he social
conce n in he i m’s objec i e ends o a ou he eme gence o he no-commi men scena io
i he ne wo k ex e nali y is posi i e and he commi men scena io i he ne wo k ex e nali y
is nega i e. This is because i ms p oduce mo e unde C( esp. NC) han unde NC ( esp.
C) i he ne wo k ex e nali y is posi i e ( esp. nega i e). Fo any gi en alue o he posi i e
ne wo k s eng h, he mo e i ms a e socially o ien ed, he la ge he inc ease in he ou pu
hey would ge by choosing o commi in ad ance o an announced ou pu le el and he
lowe he p ice consume s a e willing o pay. The o me e ec inc eases p o i s. The la e
e ec educes p o i s. When he ex en o he i m social conce n is high enough, he nega i e
p ice e ec domina es he posi i e quan i y e ec and p o i s unde C educe, hus a ou ing
he eme gence o NC as a dominan s a egy leading o a Pa e o e icien SPNE in which
sel -in e es and mu ual bene i o ou pu non-commi men do no con lic unde posi i e
ne wo k e ec s. Unlike his, o any gi en alue o he nega i e ne wo k s eng h, he mo e
i ms a e socially o ien ed, he la ge he inc ease in he ou pu hey would ge by choosing
no o commi in ad ance o an announced ou pu le el and he lowe he p ice consume s
a e willing o pay. The o me e ec inc eases p o i s. The la e e ec educes p o i s. When
he ex en o he i m social conce n is high enough, he nega i e p ice e ec domina es he
posi i e quan i y e ec and p o i s unde NC educe, hus a ou ing he eme gence o C
as a dominan s a egy leading o a Pa e o e icien SPNE in which sel -in e es and mu ual
bene i o ou pu commi men do no con lic unde nega i e ne wo k e ec s.
To sum up, he mo e i ms a e socially o ien ed he mo e he CSR-CDG shows sha p
di e en SPNE ou comes han he CDG wi h p o i -maximising i ms. The main di e ences
a e epo ed below by dis inguishing be ween posi i e and nega i e ne wo k e ec s.
1.1 Posi i e ne wo k e ec s
•I he bandwagon e ec is high enough, he SPNE would be he Pa e o e icien (C,C),
wi h he CSR-CDG being an an i-p isone ’s dilemma in which sel -in e es and mu ual
bene i o ou pu commi men do no con lic i he i m social conce n is su icien ly
low. An inc ease in he i m social conce n allows o he eme gence o wo asymme ic,
Pa e o e icien SPNE, (C,NC)and (NC,C). The e o e, he CSR-CDG becomes an an i-
coo dina ion game wi h inde e minacy ins ead o an an i-p isone ’s dilemma wi h a unique
SPNE in which sel -in e es and mu ual bene i o ou pu commi men do no con lic .
•I he bandwagon e ec is lowe , he SPNE would be he Pa e o ine icien (C,C) wi h he
CSR-CDG being a p isone ’s dilemma in which sel -in e es and mu ual bene i o ou pu
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Annals o Ope a ions Resea ch
commi men con lic i he i m social conce n is su icien ly low. An inc ease in he i m
social conce n changes he pa adigm o he game by allowing o he eme gence o (C,C)
as he unique Pa e o e icien SPNE, so ha he CSR-CDG becomes an an i-p isone ’s
dilemma in which sel -in e es andCmu ual bene i o ou pu non-commi men do no
con lic .
1.2 Nega i e ne wo k e ec s
•The SPNE would be he Pa e o ine icien (NC,NC) wi h he CSR-CDG being a p isone ’s
dilemma in which sel -in e es and mu ual bene i o ou pu non-commi men con lic i he
i m social conce n is su icien ly low. An inc ease in he i m social conce n changes he
pa adigm o he game by allowing o he eme gence o (C,C) as he unique Pa e o e icien
SPNE, so ha he CSR-CDG becomes an an i-p isone ’s dilemma in which sel -in e es
and mu ual bene i o ou pu commi men do no con lic .
We pinpoin ha choosing s a egically Co NC is no me e heo e ical specula ion.
Focusing on ne wo k indus ies, companies can make c edible announcemen s abou hei
p oduc ion le els. Fo ins ance, in Sep embe 2020, Wis on, one o he h ee Apple op
con ac manu ac u e s in India, which assembled abou 200,000 second-gene a ion iPhone
SEs pe mon h, announced a plan o scale p oduc ion up o 400,000 a mon h by he end o
ha yea (Reu e s, 2020). Mo eo e , e en i a company does no announce p ecise ou pu
le els, consume s (and i al i ms) can in e p oduc ion using as a p oxy he ( e y o en
eleased, as a simple sea ch on he websi es o , in e alias, eu e s.com, bloombe g.com,
.com, economis .com e eal) announcemen s conce ning he alue o in es men plans
ela ed o (1) new assembly lines a he old loca ion, o (2) new ac o ies.
The ich spec um o equilib ium esul s o e s many es able implica ions o econome-
icians. Fo example, i should be ound ha whe he i ms a e (1) sligh ly socially o ien ed
hey should publicly announce he ou pu o which hey aim a commi ing, and (2) highly
socially o ien ed hey should no commi o an announced ou pu le el when he ex en o
he ne wo k ex e nali y o p oduc s is small o hey should commi when he ex en o he
ne wo k ex e nali y o p oduc s is la ge.
Be o e p oceeding u he , i may be con enien o cla i y he logical iming o he e en s,
which ep esen s a heo e ical modelling syn hesis in a s a ic one-sho en i onmen o he
his o ical iming o he e en s o he CSR-CDG, which in u n a ec s he beha iou o
consume s and i ms. The logical s uc u e used in he p esen wo k esembles hose adop ed
by Choi & Lim (2022), desc ibing he CDG wi h p o i -maximising i ms ha compe e
simul aneously à la Cou no in he ma ke s age o he game. This is because he eade may
wonde whe he he s a egic use o ou pu commi men in a ne wo k indus y a ec s he
consume s’ expec a ions2and he na u e o he compe i ion in he ou pu ma ke .
Fi s , we pinpoin ha becoming a S ackelbe g leade is no necessa ily ela ed o playing
Co NC a he decision-making s age o he CDG, bu a he because one o he wo i ms
2The expec a ions o ma ion mechanisms ha e been ex ensi ely deba ed in he economic li e a u e. The
subjec is ex emely ele an in di e en kinds o dynamic models o ackle he issue o (exogenous o
endogenous) luc ua ions. This has been done, e.g., in cobweb models (Go i e al., 2015; Hommes, 1994), OLG
models (Fan i & Go i, 2013; Go i & Sodini, 2020,2021) o dynamic oligopolies (Bischi e al., 1998,1999). In
a con ex o a s a ic oligopoly, consume s’ expec a ions become ele an in ne wo k indus ies. In his ega d,
some ecen con ibu ions poin ed his ques ion ou o he a en ion, e.g., G i a & Ve as (2011), Suleymano a
&Wey(2012), Hagiu & Halabu da (2014), Toshimi su (2019,2021), Leppänen (2020), Nakamu a (2021).
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Annals o Ope a ions Resea ch
can o canno p oduce be o e he i al in he logical iming o e en s.3In he i s case,
i becomes he leade a he ma ke s age o he game, wi h he i al p oducing only a e
obse ing he i al’s ou pu . In he second case, i ms p oduce simul aneously, and each
i m is only able o expec he i al’s p oduc ion. The CSR-CDG can e en ually be played
sequen ially (S ackelbe g) o simul aneously (Cou no ) a he ma ke s age. The comple e
logical sequence o e en s in each o he wo cases is as ollows.
•Cou no compe i ion in he ma ke s age o he CDG. In he i s decision-making s age,
each i m chooses o c edibly commi o no o commi o an announced ou pu le el. This
is a bina y choice ha happens be o e consume s make hei pu chase decisions and hen
a ec s hei expec a ions abou he p e ailing ne wo k size. Consume s a e a ional and
ha e a ional expec a ions. I a leas one o he wo i ms is commi ing, he consume s’
expec a ions abou he p oduc ion o he commi ing i m(s) a e ealised immedia ely, i.e.,
a he same logical ime as he announcemen wi hou he need o know he p e ailing
equilib ium. I a leas one o he wo i ms is no commi ing, consume s’ expec a ions
become binding, and hey a e ealised in equilib ium as consume s a e a ional so ha hey
can know in ad ance (in he logical iming o he e en s) he p e ailing equilib ium alue
o he ou pu . This hold in an in e media e s age be ween he decision-making s age and
he ma ke s age. In he ma ke s age, compe i ion be ween i ms holds simul aneously,
i.e., each o hem maximises i s objec i e gi en he expec a ions abou he p oduc ion
o he i al. Indeed, acco ding o he Cou no ules, ma ke compe i ion in he his o ical
iming o he e en s can be ep esen ed by using a model in which i ms make p oduc ion
decisions simul aneously. In his case, each i m has expec a ions abou he p oduc ion o
he i al. These expec a ions will be ealised in equilib ium (i.e., when he ou pu eac ion
unc ion c oss each o he ). We pinpoin ha he non-commi men scena io is equi alen
o he case in which i ms a e commi ing bu hei announcemen is no c edible. In his
case, consume s beha e as i hey o med expec a ions be o e he commi men and hey
a e sel - ul illing, in u n, esponding o equilib ium quan i y. “Compa ed o he case o
commi men , his implies ha a low ou pu le el a ises in he ne wo k ma ke ” (Choi &
Lim, 2022, p. 665) when he ne wo k ex e nali y is posi i e.
•S ackelbe g compe i ion in he ma ke s age o he CDG. The iming o he e en s o
he i ms’ beha iou in he i s decision-making s age and he consume s’ beha iou
exac ly eplica e hose de ailed in he p e ious poin . In he ma ke s age, compe i ion
be ween i ms holds sequen ially (ins ead o simul aneously), i.e., one i m mo es i s
and becomes he leade ; he i al chooses i s p oduc ion a e obse ing he p oduc ion
o he leade and hen becomes he ollowe . The leade conside s he ou pu eac ion
unc ion o he ollowe . Indeed, acco ding o he S ackelbe g ules, ma ke compe i ion
in he his o ical iming o he e en s can be ep esen ed by using a model in which i ms
make p oduc ion decisions sequen ially, bu he choice abou ou pu commi men does no
a ec he kind o compe i ion in he ou pu ma ke . This implies ha he commi ing i m
can be he leade o he ollowe . The s uc u e o he ou sub-games o he CSR CDR à la
S ackelbe g (which is epo ed in he appendix) is he ollowing: i bo h i ms a e ( esp. a e
no ) commi ing, he leade chooses o commi ( esp. no o commi ) be o e he ollowe .
3An anonymous e iewe pinpoin s some doub s on he ac i m ipublicly announces i s quan i y, bu he
i al, i m j, seems igno ing his in o ma ion, and hen sugges s conside ing S ackelbe g compe i ion when
only one i m is unila e ally commi ing ( he C- i m is he leade and he NC- i m is he ollowe ). In he las
sec ion o he Appendix, we explain he easons why we belie e ha bo h app oaches a e logically consis en
and de elop he CSR CDG by conside ing S ackelbe g compe i ion in he asymme ic sub-games C/NC and
NC/C. Rema kably, also unde his assump ion he SPNE solu ions o he game a e quali a i ely simila .
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I he leade is commi ing and he ollowe does no , he leade chooses o play Cbe o e
he ollowe makes p oduc ion decisions by playing NC. I he leade is no commi ing
and he ollowe is commi ing, he leade chooses o play NC be o e he ollowe makes
p oduc ion decisions by playing C.
The a icle p oceeds as ollows. Sec ion 2p esen s he basic elemen s o he non-
coope a i e CSR-CDG based on Cou no i al y. Sec ion 3( esp. 4) s udies he endogenous
ma ke con igu a ion eme ging in his en i onmen by conside ing p o i s ( he i m’s u il-
i y) as he main decision a iable in he decision-making s age. Sec ion 5ou lines he main
conclusions. The Appendix shows (i) he mic oeconomic ounda ions o he ma ke demand
unde Cand NC, (ii) some analy ical de ails o he model p esen ed in he main ex , (iii) he
CSR-CDG in which he s a egic in e ac ion, in he ma ke s age o he game, be ween he
wo i ms occu s sequen ially (S ackelbe g) ins ead o simul aneously (Cou no ) by assuming
ha i m iis always he leade , (i ) he CSR-CDG à la Cou no -S ackelbe g in which he
commi ing i m is always he leade , and ( ) a compa ison be ween Cou no and S ackelbe g
scena ios showing, depending on he pa ame ic se , ha bo h kinds o compe i ion se ings
can eme ge endogenously.
2Themodel
Conside a Cou no duopoly in which i ms p oduce homogeneous ne wo k goods (Ka z &
Shapi o, 1985). The ne wo k e ec (consump ion ex e nali y) can be posi i e (e.g., mobile
communica ions, so wa e, in e ne - ela ed ac i i ies, online social ne wo ks, ashion, e c.)
o nega i e (e.g., a ic conges ion o ne wo k conges ion o e limi ed bandwid h). Unde a
posi i e ( esp. nega i e) ex e nali y an inc easing numbe o use s inc eases ( esp. educes)
he indi idual u ili y and he alue o he goods o each consume , in u n, causing he so-
called bandwagon ( esp. snob) e ec . As in Ka z & Shapi o (1985), he e exis wo pola
cases: (1) i ms can commi c edibly hemsel es o an announced ou pu le el, gi en he
ne wo k size, be o e consume s maximise hei u ili ies (C); (2) i ms canno (a e unable o)
commi hemsel es o a gi en ou pu le el, so ha consume s o m hei expec a ions on o al
sales, which a e ul illed in equilib ium acco ding o he a ional expec a ions hypo hesis
(NC). These wo al e na i es ha e been aken exogenously in Ka z & Shapi o (1985)and
endogeneised by Choi & Lim (2022), who build on he non-coope a i e CDG wi h comple e
in o ma ion p o i -maximising i ms.
The main aim o he p esen a icle is o add he decision-making s age o a Cou no
duopoly à la Ka z & Shapi o (1985) and ollows Choi & Lim (2022) by assuming socially
o ien ed, ins ead o p o i -maximising, i ms, in line wi h he aim o mode n i ms amed in
s a egic compe i i e ma ke s (as clea ly epo ed by McKinsey, 2019, and discussed in he
in oduc ion), and building on he non-coope a i e CSR-CDG.
The mic oeconomic ounda ions o he (linea ) ma ke demand wi h ne wo k ex e nali y
and ull p oduc compa ibili y ollow he ecen a icle by Buccella e al. (2022a), p o iding
de ails ( epo ed in he appendix) o how he commi men scena io and he non-commi ing
scena io a ec he demand o consume s o he p oduc o ne wo k i.
Unde he al e na i e C o bo h i ms, he (no malised) in e se ma ke demand o i m i
is:
pC/C
i=1−qi−qj+n(qi+qj)(1)
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Annals o Ope a ions Resea ch
whe e pC/C
iis he ma ginal willingness o pay owa ds p oduc s o ne wo k i(i,j={1,2},
i= j) unde ou pu commi men , qideno es he quan i y o he goods p oduced by i m i
and n∈(−1,1)is he s eng h o he ne wo k e ec ( he highe he absolu e alue o n, he
s onge he e ec o he ne wo k). Posi i e ( esp. nega i e) alues o n e e s o he case o
posi i e ( esp. nega i e) ne wo k goods.
Unde he al e na i e NC o bo h i ms, he in e se demand unc ions (see, e.g., Hoe nig,
2012; Chi co & Sc imi o e, 2013; Bha acha jee & Pal, 2014; Fan i & Buccella, 2016;Buc-
cella e al., 2022a) a e as ollows:
pNC/NC
i=1−qi−qj+n(yi+yj)(2)
whe e yi(i,j={1,2},i= j) deno es he consume s’ expec a ions abou he equilib ium
ou pu p oduced by i m i,andpNC/NC
iis he ma ginal willingness o pay owa ds p oduc s
o ne wo k iunde ou pu non-commi men .4
The gene ic i m i’s p o i unc ion is gi en by:
i=(pi−w)qi(3)
whe e 0 ≤w<1 is he cons an a e age and ma ginal cos (i.e., he echnology has cons an
e u ns o scale), which is se o ze o o analy ical ac abili y and wi hou loss o gene ali y.
The e o e, he p o i unc ions in he symme ic scena ios C/Cand NC/NC ead espec-
i ely as ollows5:
C/C
i=[1−qi−qj+n(qi+qj)]qi(4)
and
NC/NC
i=[1−qi−qj+n(yi+yj)]qi(5)
Following ecen ly es ablished li e a u e (e.g., Goe ing, 2007,2008;Lambe ini&
Tampie i, 2010,2015; Kopel & B and, 2012;Kopele al.,2014; Lambe ini e al., 2016;
Fan i & Buccella, 2017,2018; Kopel & Laman ia, 2018; Plane -F ied ich & Sahm, 2020),6
we assume ha he social conce ns can be in e p e ed as pa o he consume su plus (CS).
This, in u n, implies ha he ea u e o a CSR i m is o be sensi i e owa ds CS. The e o e,
each i m aims a maximising a weigh ed sum o p o i s ( ha acc ue o i s owne s) and he
consume su plus ( ha acc ues o i s s akeholde s). Mo e o mally, we de ine bas he pe cen -
age o he consume su plus cap u ed by he s akeholde s. I he s akeholde s ully cap u e
he su plus, hen b=1. I none o he consume su plus acc ues o he s akeholde s, hen
b=0 and he i m’s objec i e boils down o p o i maximisa ion. The inclusion o a ac ion
o he wel a e o consume s can also be hough o as being pa o he i m’s social conce n
o ca e o consume ou comes. Then, each i m aims a weigh ing he consume su plus
wi h b∈(0,1), which is he gene al case. This pa ame e can be exogenous o endogenous.
4Fo he sake o comple eness, we epo he demand o he p oduc o ne wo k iin he asymme ic scena io
in which only one i m, say i m i, c edibly commi s i sel o an announced ou pu le el and he i al, say i m j,
doesno (de ailsa e epo edin heappendix).Thisisgi enby heexp ession pC/NC
i=1−qi−qj+n(qi+yj).
This is done by assuming ha consume s a e homogeneous and he C- i m neglec s he in luence on NC- i m’s
consume s.
5The p o i unc ions o he commi ing i m iand he non-commi ing i m jin he asymme ic scena io
C/NC a e espec i ely gi en by he ollowing exp essions C/NC
i=[1−qi−qj+n(qi+yj)]qiand
C/NC
j=[1−qi−qj+n(qi+yj)]qj.
6On he empi ical side, see Siegel & Vi aliano (2007) and Fe nández-K anz & San aló (2010), who p o ide
some empi ical indings abou CSR.
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Annals o Ope a ions Resea ch
Bo h cases ha e been analysed in he li e a u e (Fan i & Buccella, 2017; Plane -F ied ich &
Sahm, 2020) especially in models dealing wi h CSR decision games ha con as he bina y
choice CSR e sus p o i maximisa ion (PM). Howe e , his is no he main aim o he p esen
a icle as all he i ms a e CSR-o ien ed and mus choose o play Co NC ( he commi men
decision game). In his wo k, he e o e, we assume ha pa ame e bis aken exogenously by
each i m.
The CSR-o ien ed i m i’s objec i e (u ili y) unc ion Uimay be speci ied as a simple
pa ame e ised combina ion o p o i s and consume su plus, and i is gi en by:
UC/C
i=C/C
i+bCSC/C(6)
i bo h i ms c edibly commi hemsel es o an announced ou pu le el, and
UNC/NC
i=NC/NC
i+bCSNC/NC (7)
i bo h i ms do no commi hemsel es o an announced ou pu le el, whe e.7
CSC/C=1
2(1−n)(qi+qj)2(8)
and
CSNC/NC =1
2(qi+qj)[qi+qj−n(yi+yj)](9)
In he i s (decision) s age o his non-coope a i e game wi h comple e in o ma ion, each
i m chooses o commi o no o commi hemsel es o an announced ou pu le el. This
ep esen s a s anda d bina y choice, which can be based ei he on p o i s (i he decision
is owne s-o ien ed) o on he i m’s u ili y (i he decision is s akeholde s-o ien ed). Bo h
objec i es can be jus i ied bu ollow a di e en na a i e. The a ionale o conside p o i s
as he main decision a iable in he i s decision-making s age implies, ollowing B and and
G o he (2015), ha i ms aims a emaining ope a i e in he ma ke o e ime so ha hei
p o i s mus be non-nega i e (see also F iedman, 1970, who poin ed ou ha he maximisa ion
o p o i s o sha eholde s is he only social esponsibili y o businesses).
The a ionale o conside a b oade objec i e (i.e., he i m’s u ili y) as a s a egic a iable
is mo e in line wi h he main goal o a s akeholde s-o ien ed i m and ollows, e.g., Buccella
e al. (2022b). In his ega d, Kopel e al., (2014, p. 397) p o ide a nice in e p e a ion o jus i y
he s a egic use o he i m’s u ili y: a combined objec i e o p o i s and weigh ed consume
su plus aims a cap u ing “a i m ype e e ed o as sync e ic s ewa dship model…, e e ing
o an o ganiza ion which emb aces economic as well as non-economic goals. Likewise, we
can hink o a hyb id o ganiza ional s uc u e, which enables he i m o pu sue p o i and
non-p o i mo i es”.
In he second (ma ke ) s age, each i m compe es à la Cou no in he p oduc ma ke and
hen chooses he quan i y o maximise he i m’s u ili y U.
2.1 Fi ms commi hemsel es o an announced ou pu le el
Conside he symme ic sub-game in which bo h i ms commi hemsel es o an announced
ou pu le el (C/C). Gi en he CSR i m’s objec i e and hen conside ing he exp essions in
7In he asymme ic scena io C/NC, i mi’s and i m j’s objec i es a e espec i ely gi en by UC/NC
i=
C/NC
i+bCSC/NC and UC/NC
j=C/NC
j+bCSC/NC, whe e he consume su plus CSC/NC =
1
2(qi+qj)[qi+qj−n(qi+yj)].
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Annals o Ope a ions Resea ch
unique Pa e o ine icien SPNE and he CSR-CDG is a p isone ’s dilemma o any
nA(b)<n<nC(b), in which sel -in e es and mu ual bene i o ou pu commi men
con lic ; (2.4) (C,C)is he unique Pa e o e icien SPNE and he CSR-CDG is an
an i-p isone ’s dilemma (deadlock) o any nC(b)<n<1, in which sel -in e es and
mu ual bene i o ou pu commi men do no con lic .
[3] I 0.313 <b<0.393 hen (3.1) (NC,NC)is he unique Pa e o e icien SPNE
and he CSR-CDG is an an i-p isone ’s dilemma (deadlock) o any 0 ≤n<nA(b),
in which sel -in e es and mu ual bene i o ou pu non-commi men do no con lic ;
(3.2) (C,NC)and (NC,C)a e wo asymme ic SPNE, and he CSR-CDG is an an i-
coo dina ion game o any nA(b)<n<nB(b); (3.3) (C,C)is he unique Pa e o
ine icien SPNE and he CSR-CDG is a p isone ’s dilemma o any nB(b)<n<
nC(b), in which sel -in e es and mu ual bene i o ou pu commi men con lic ; (3.4)
(C,C)is he unique Pa e o e icien SPNE and he CSR-CDG is an an i-p isone ’s
dilemma (deadlock) o any nC(b)<n<1, in which sel -in e es and mu ual bene i
o ou pu commi men do no con lic .
[4] I 0.393 <b<0.4013 hen (4.1) (NC,NC)is he unique Pa e o e icien SPNE
and he CSR-CDG is an an i-p isone ’s dilemma (deadlock) o any 0 ≤n<nA(b),
in which sel -in e es and mu ual bene i o ou pu non-commi men do no con lic ;
(4.2) (C,NC)and (NC,C)a e wo asymme ic SPNE, and he CSR-CDG is an an i-
coo dina ion game o any nA(b)<n<nB(b); (4.3) (C,C)is he unique Pa e o
e icien SPNE and he CSR-CDG is an an i-p isone ’s dilemma (deadlock) o any
nB(b)<n<1, in which sel -in e es and mu ual bene i o ou pu commi men do
no con lic .
[5] I 0.4013 <b<0.5 hen (5.1) (NC,NC)is he unique Pa e o e icien SPNE and
he CSR-CDG is an an i-p isone ’s dilemma (deadlock) o any 0 ≤n<nA(b),
in which sel -in e es and mu ual bene i o ou pu non-commi men do no con lic ;
(5.2) (C,NC)and (NC,C)a e wo asymme ic SPNE, and he CSR-CDG is an an i-
coo dina ion game o any nA(b)<n<1.
Le nbe nega i e.
[6] I 0 ≤b<0.115 hen (NC,NC)is he unique Pa e o ine icien SPNE and he CSR-
CDG is a p isone ’s dilemma o any 0 >n>−1, in which sel -in e es and mu ual
bene i o ou pu non-commi men con lic .
[7] I 0.115 <b<0.168 hen (7.1) (NC,NC)is he unique Pa e o ine icien SPNE and
he CSR-CDG is a p isone ’s dilemma o any 0 >n>−nB(b), in which sel -in e es
and mu ual bene i o ou pu non-commi men con lic ; (7.2) (C,C)and(NC,NC)a e
wo symme ic SPNE (Cpayo domina es NC) and he CSR-CDG is a coo dina ion
game o any −nB(b)>n>−1.
[8] I 0.168 <b<0.219 hen (8.1) (NC,NC)is he unique Pa e o ine icien SPNE
and he CSR-CDG is a p isone ’s dilemma o any 0 >n>−nB(b),inwhich
sel -in e es and mu ual bene i o ou pu non-commi men con lic ; (8.2) (C,C)and
(NC,NC)a e wo symme ic SPNE (Cpayo domina es NC) and he CSR-CDG is a
coo dina ion game o any −nB(b)>n>−nA(b); (8.3) (C,C)is he unique Pa e o
e icien SPNE and he CSR-CDG is an an i-p isone ’s dilemma (deadlock) o any
−nA(b)>n>−1, in which sel -in e es and mu ual bene i o ou pu commi men
do no con lic .
[9] I 0.219 <b<0.333 hen (C,C)is he unique Pa e o e icien SPNE and he CSR-
CDG is an an i-p isone ’s dilemma (deadlock) o any 0 >n>−1, in which sel -
in e es and mu ual bene i o ou pu commi men do no con lic .
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Annals o Ope a ions Resea ch
[10] I 0.333 <b<bNC/NC
T(n) hen (C,C)is he unique Pa e o e icien SPNE and he
CSR-CDG is an an i-p isone ’s dilemma (deadlock) o any 0 >n>−1−2b
b,inwhich
sel -in e es and mu ual bene i o ou pu commi men do no con lic .
P oo . Appendix.
The economic in ui ion behind he esul s o P oposi ion 1 is simple and p oceeds ollowing
he s anda d mechanism o p ice/quan i y on he i m’s p o i . The esul s a e d i en by he
in e ac ion be ween he ne wo k size and he deg ee o he i m’s social conce n (Fig. 1).
I i ms a e p o i -maximising en i ies (b=0), he model eplica es he ou comes o Choi
&Lim(2022), in which he commi men scena io eme ges when he ne wo k ex e nali y is
posi i e (commi ing o an announced ou pu le el is a ou ed by he bandwagon e ec o he
posi i e ne wo k ex e nali ies), and he no-commi men scena io eme ges when he ne wo k
ex e nali y is nega i e (no commi ing o an announced ou pu le el is a ou ed by he snob
e ec o he nega i e ne wo k ex e nali ies).
Unlike his case, i i ms a e CSR-o ien ed and he ne wo k ex e nali y is posi i e, an
inc ease in he deg ee o social conce n sha ply changes he SPNE and hen he wel a e
ou comes (Fig. 2, Panels A-B), wo king ou a ou ing (i) he no commi men scena io (which
is Pa e o e icien o i ms bu no consume s o ien ed) i he ex en o he ne wo k ex e nali y
is ela i ely small, (ii) he mixed scena io, in which only i m aims a commi ing o an
announced ou pu le el, bu he i al does no i he ex en o he ne wo k ex e nali y becomes
la ge . This ou come also leads o he highes social wel a e, al hough i does no ep esen
Fig. 1 The CSR-CDG: SPNE when nand b a y and he i m’s p o i is he decision a iable. The sand-colou ed
egion ep esen he un easible pa ame e space, which is bounded by he h eshold b<1
2−n:= bNC/NC
T(n).
The easibili y condi ion mus be sa is ied o gua an ee posi i e p o i s in he sub-game NC/NC. Yellow
egion: (C,C)and (NC,NC)a e wo symme ic SPNE (NC payo domina es C) and he CSR-CDG is a
coo dina ion game. G een egion: (C,C)and (NC,NC)a e wo symme ic SPNE (Cpayo domina es NC)
and he CSR-CDG is a coo dina ion game. Ligh -blue egion: (C,NC)and (NC,C)a e wo asymme ic
SPNE wi h he C- i m ge ing he bes ou come and he CSR-CDG is an an i-coo dina ion game. This acco ds
wi h he egion below, in which he CSR-CDG is an an i-coo dina ion game and C- i m con inues o ge he
bes ou come
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Annals o Ope a ions Resea ch
Fig. 2 Equilib ium social wel a e as a unc ion o b. The black ( esp. ed) [ esp. blue] cu e e e s o he case
C/C( esp. NC/NC) [ esp. C/NC]. The solid lines ep esen he social wel a e p e ailing a he SPNE.
The dash-do ed lines ep esen s he social wel a e le el ha can eme ging when he e is inde e minacy, i.e., a
mul iplici y o SPNE in pu e s a egies and he CSR-CDG is a coo dina ion game. The do ed lines a e ic i ious
and a e d awn only o compa ison pu poses. Panel A: n=0.8; Panel B: n=0.4; Panel C: n=−0.8
a win–win solu ion om a socie al pe spec i e as consume s would be be e o by unde
C/C. The ex en o social conce n, he e o e, a ou s he eme gence o he no-commi men
scena io. This is because an inc ease in binc eases he p oduc ion o he i ms in all he
s a egic p o iles, bu he pe cen age inc ease in he ou pu o he C- i ms is la ge han he
pe cen age inc ease in he ou pu o he NC- i ms. The e o e, he p o i s o he C- i m educe
mo e han he p o i s o he NC- i m and his, in u n, ends o le NC become he dominan
s a egy i he ne wo k ex e nali y is no oola ge. When he ex en o he ex e nali y becomes
la ge he e ec s o an inc ease in ba e di e en . Fi s , he e is a s anda d posi i e e ec
o p oduc ion in all he s a egic p o iles and his, in u n, educes he co esponding ma ke
p ice. Howe e , he pe cen age inc ease in ou pu o he C- i m is la ge han he pe cen age
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Annals o Ope a ions Resea ch
inc easeinou pu o he NC- i m.Inaddi ion, he educ ionin hema ke p icewhen i msa e
commi ing is la ge han when i ms a e no commi ing. This, in u n, leads o a educ ion in
p o i sunde Candaninc easeinp o i sunde NCwi h heeme genceo anan i-coo dina ion
game, in which each i m plays he opposi e s a egy o he i al in equilib ium.
Things change unde nega i e ne wo k ex e nali ies (Figs. 1and 2, Panel C). In his case,
an inc ease in he ex en o he social conce n wo ks ou in he di ec ion o inc easing he
impo ance o he snob e ec , so ha an inc ease in he (absolu e alue o ) s eng h o he
ne wo k ex e nali y ends o shi inwa ds he ma ke demand a he highes in ensi y. This
has nega i e e ec s on p o i s in e e y s a egic p o ile. Howe e , he educ ion in ou pu
expe ienced unde Cis la ge han unde NC le ing he no-commi men scena io become
he dominan s a egy. Howe e , as i ms unde NC p oduce mo e (and sell a a lowe p ice)
han unde C he e is a join incen i e o de ia e owa ds he commi men scena io, bu no
one has he unila e al incen i e o comply wi h he (non-coope a i e) ag eemen , so each
i m is en apped in he po e y ap o do no commi o an announced ou pu le el. As he
NC- i m p oduces mo e han he C- i m when nis nega i e, an inc ease in he ex en o b
has exac ly he opposi e e ec s as hose desc ibed abo e so ha he educ ion in p o i s o he
NC- i m is la ge han he educ ion in p o i s o he C- i m and Cbecomes he dominan
s a egy o la ge alues o b. As consume s a e now always be e o unde NC he e a e
no win–win solu ions also in his case.
The policy ecipes a e pe haps qui e a icula ed ( he e a e no win–win si ua ions, in which
bo h consume s and i ms a e be e o ) and coun e in ui i e. I he ne wo k ex e nali y
is posi i e, inc easing he impo ance o he social conce n a ou s he eme gence o no-
commi men scena io, which is de imen al o consume s and social wel a e. The e o e, i
he ex en o he ne wo k ex e nali y is ela i ely high, he public au ho i y should no a ou
he eme gence o a c edible commi men o all he i ms as he bes ou come a he socie al
le el is a si ua ion in which one i m commi s o an announced ou pu le el, bu he i al
does no ( hough consume s would be be e o unde he ull commi men scena io), and
he ex en o he social conce n should be se an in e media e le el; i he ex en o he
ne wo k ex e nali y is ela i ely low, he public au ho i y should a ou he eme gence o
a c edible commi men o all he i ms ( hough he socie y would be be e o unde he
mixed scena io), bu inc easing he impo ance o he social conce n is de imen al o i ms.
I he ne wo k ex e nali y is nega i e, inc easing he impo ance o he social conce n a ou s
he eme gence o he commi men scena io, which is de imen al o consume s and social
wel a e as socie y would be be e o unde no commi men . The e o e, he ex en o he
social conce n should be se a an in e media e le el.
3.2 S akeholde s’u ili y as he decision a iable in he i s decision-making s age
o he CSR-CDG
Unlike Sec . 3.1, his sec ion examines he i s s age o he game by assuming ha he choice
o commi o no commi is s akeholde -o ien ed and hen he i m’s u ili y ep esen s he
decision a iable. In his ega d, Table 2summa ises he payo ma ix wi h he i m’s u ili y
unc ions in he symme ic and asymme ic sub-games (see Eqs. (14), (21), (35)and(36)).
To de i e all he possible equilib ia o he game one mus s udy he sign o he i m’s
u ili y di e en ials o i={1,2},i= j, ha is:
UA(n,b):= U∗C/NC
i(n,b)−U∗NC/NC
i(n,b)(44)
UB(n,b):= U∗NC/C
i(n,b)−U∗C/C
i(n,b)(45)
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Annals o Ope a ions Resea ch
Table 2 The CSR-CDG à la Cou no (payo ma ix). Fi m’s u ili y
Fi m j→CNC
Fi m i↓
CU
∗C/C
i(n,b), U∗C/C
j(n,b)U∗C/NC
i(n,b), U∗C/NC
j(n,b)
NC U∗NC/C
i(n,b), U∗NC/C
j(n,b)U∗NC/NC
i(n,b), U∗NC/NC
j(n,b)
and
UC(n,b):= U∗NC/NC
i(n,b)−U∗C/C
i(n,b)(46)
F om (44), he sign o UA(n,b)is posi i e ( esp. nega i e) i n>nU
A(b)( esp. n<
nU
A(b),whe enU
A(b)is he h eshold alue o he ex en o he ne wo k ex e nali y (as a
unc ion o he ex en o he i m’s social conce n) such ha UA(n,b)=0. We do no
epo he alue o nU
A(b)as i is analy ically cumbe some and no in o ma i e.
F om (45), he sign o UB(n,b)is nega i e ( esp. posi i e) i n>nU
B(b)( esp. n<
nU
B(b),whe e
nU
B(b):= (3−2b)(1−2b)
b2−b−1(47)
is he h eshold alue o he ex en o he ne wo k ex e nali y (as a unc ion o he ex en o
he i m’s social conce n) such ha UB(n,b)=0.
F om (46), he sign o UC(n,b)is nega i e ( esp. posi i e) i n>nU
C(b)( esp. n<
nU
C(b),whe e
nU
C(b):= (1+2b)(3−2b)
4−b(9−4b)(48)
is he h eshold alue o he ex en o he ne wo k ex e nali y (as a unc ion o he ex en o
he i m’s social conce n) such ha UC(n,b)=0.
To a oid leng hening he a icle u he we do no w i e an explici p oposi ion o de ail
he SPNE ou comes and he endogenous ma ke s uc u e o he CSR-CDG eme ging when
he i m’s u ili y is he decision a iable. Howe e , wi hou loss o gene ali y, esul s a e
epo ed Fig. 3, which esembles Fig. 1abo e. I is in e es ing o obse e ha , as expec ed,
when he ne wo k ex e nali y is posi i e and he decision is s akeholde -o ien ed in he i s
s age o he CSR-CDG, p oduc ion weigh s mo e in he i m’s objec i e han when p o i is
he decision a iable, and his a ou he eme gence o he commi men scena io i espec i e
o he le el o he social conce n, in sha p con as wi h he p e ious case. This “opposi e
ou come” is also obse ed when he ne wo k ex e nali y is nega i e. In his case, in ac , he
no-commi men scena io eme ges o a wide pa ame e con igu a ion han when p o i is he
decision a iable. This is because, in his case, p oduc ion is highe unde no commi men
han unde commi men .
123
Annals o Ope a ions Resea ch
Fig. 3 TheCSR-CDG: SPNEwhen nandb a yand he i m’su ili y is hedecision a iable.Thesand-colou ed
egion ep esen he un easible pa ame e space, which is bounded by he h eshold b<1
2−n:= bNC/NC
T(n).
The easibili ycondi ionmus be sa is ied o gua an ee posi i ep o i sin he sub-game NC/NC. G een egion:
(C,C)and(NC,NC)a e wo symme ic SPNE (Cpayo domina es NC) and he CSR-CDG is a coo dina ion
game. Ligh -blue egion: (C,NC)and (NC,C)a e wo asymme ic SPNE wi h he NC- i m ge ing he bes
ou come and he CSR-CDG is an an i-coo dina ion game
4 Conclusions
The inc easingly apid sp ead o ne wo k goods, which is making consump ion ex e nali ies
a majo issue, and Co po a e Social Responsibili y, which is making socially o ien ed i ms
mo e and mo e impo an in he ma ke , is becoming a s ylised ac especially in s a egic
compe i i e con ex s.
Unde ne wo k consump ion ex e nali ies, consume s mus ei he o m ( a ional) expec-
a ions o know he in o ma ion disclosed by he i ms abou he size o compe ing ne wo ks.
This a icle aimed a di ec ly ex ending Ka z & Shapi o (1985) and Choi & Lim (2022)
by conside ing CSR ins ead o p o i -maximising i ms. In doing his, i endogeneised he
choice o c edibly commi o an announced ou pu le el, gi en he ne wo k size, be o e con-
sume s make hei pu chasing decisions. The a icle hen conside s ou pu commi men as
a s a egic de ice ha depends on he ex en o he ne wo k size and he deg ee o i ms’
social conce n cap u ing he impo ance o consume wel a e in p oduc ion. Unlike Choi &
Lim (2022), who s udy he non-coope a i e, mul i-s age, commi men decision game wi h
p o i -maximising i ms, he p esen wo k conside s he non-coope a i e, mul i-s age, CSR
commi men decision game.
Ou non-coope a i e game allows us o obse e se e al equilib ium scena ios, in sha p
con as o Choi & Lim (2022), anging om he p isone ’s dilemma o he an i-p isone ’s
dilemma (deadlock), passing h ough he an i-coo dina ion game and o e s some es able
implica ions o econome icians abou he ma ke con igu a ion ha can ha e ele an wel a e
e ec s, which a e d i en by he ex en o he i m’s social conce n.
123

Annals o Ope a ions Resea ch
The p esen wo k belongs o an eme ging indus ial o ganiza ion li e a u e (e.g., G i a &
Ve as, 2011; Hagiu & Halabu da, 2014; Leppänen, 2020; Nakamu a, 2021; Suleymano a &
Wey, 2012;Toshimi su,2019,2021)on heconsume s’expec a ionsandne wo kex e nali ies
andmaybeex endedinsomedi ec ions, o ins ance byadding R&Din es men s,manage ial
delega ion, unions and so on.
Acknowledgemen s Theau ho sacknowledge woanonymous e iewe so hejou nal o aluablecommen s
on an ea lie d a o he manusc ip . Luca Go i acknowledges inancial suppo om he Uni e si y o Pisa
unde he “PRA – P oge i di Rice ca di A eneo” (Ins i u ional Resea ch G an s) – P ojec No. PRA_2020_64
“In ec ious diseases, heal h and de elopmen : economic and legal e ec s”. Mau o Sodini acknowledges inan-
cialsuppo om heCzechScienceFounda ion(GACR)unde p ojec 23-06282S,andanSGS esea chp ojec
o VŠBTUO (SP2023/19). The usual disclaime applies. This s udy was conduc ed when Domenico Buccella
was a isi ing schola a he Depa men o Law o he Uni e si y o Pisa.
Funding Openaccess undingp o idedbyUni e si àdiPisawi hin he CRUI-CARE Ag eemen .Theau ho s
decla e ha his s udy was unded by he Uni e si y o Pisa and he Technical Uni e si y o Os a a.
Decla a ions
Con lic o in e es The au ho s decla e ha hey ha e no con lic o in e es .
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Appendix
The mic oeconomic ounda ions o he ma ke demand
when bo h i ms do no commi hemsel es o an announced ou pu
le el
This sec ion b ie ly p esen s he mic oeconomic ounda ions o he ma ke demand epo ed
in Eq. (2) in he main ex . I ackles his issue by conside ing i ms ha do no commi
hemsel es o an announced ou pu le el be o e consume s make hei pu chase decisions.
This ollows he mechanism de ailed by Ka z & Shapi o (1985) in he model p esen ed in
he main body o he ex o hei con ibu ion wi h consume s ha ing a ional expec a ions.
On he supply side, he economy is bi-sec o ial wi h a compe i i e sec o p oducing he
nume ai e good mand a duopolis ic indus y in which i m iand i m j(i={1,2};i= j)
p oduce ne wo k goods o a ie y (ne wo k) iand a ie y (ne wo k) j, espec i ely. These
goods a e pe cei ed as homogeneous by cus ome s.
On hedemand side, he e a e iden ical consume s wi h p e e ences desc ibed by he u ili y
unc ion V=Z+m, which is linea in he nume ai e good m. The u ili y Vis maximised
subjec o he budge cons ain piqi+pjqj+m=R,whe eZis a wice con inuously
di e en iable unc ion, qiand qja e he con ol a iables o he p oblem, piand pj ep esen
he p ice o (i.e., he ma ginal willingness o pay o he ep esen a i e consume o ) p oduc
o a ie y iand a ie y j, espec i ely, and Ris he consume ’s exogenous nominal income.
This income is high enough o a oid he exis ence o income e ec s on he demand o qiand
123
Annals o Ope a ions Resea ch
qj(i.e., he goods en e non-linea ly in V). In his ega d, in ac , he u ili y unc ion Vis
quasi-linea in mso ha all he ela ed p ope ies abou he demand o mand ha o qiand
qjhold (Ami e al., 2017; Choné & Linneme , 2020).
Di e en om he adi ional IO li e a u e, we assume he exis ence o ne wo k ex e nali-
iesinconsump ion,i.e.,onepe son’sdemandalsodependson hedemando o he cus ome s.
In his ega d, yi ep esen s an ex e nal e ec deno ing he consume s’ expec a ions abou
i m i’s equilib ium o al sales.
The simple mechanism o ne wo k e ec s desc ibed so a ollows Ka z & Shapi o (1985)
and esembles se e al ecen a icles belonging o he IO li e a u e amed in s a egic com-
pe i i e ma ke s using wi h linea ma ke demand (Buccella e al., 2022a,2022b;Chi co&
Sc imi o e, 2013; Choi & Lim, 2022;Hoe nig,2012; Naska & Pal, 2020;Sh i as a ,2021;
Song & Wang, 2017).
The unc ion Z ollows he speci ica ion o a quad a ic u ili y8:
Z=qi+qj−1
2(q2
i+q2
j+2qiqj)
+n[qi(yi+yj)+qj(yj+yi)]−n
2(y2
i+y2
j+2yiyj)(A.1)
whe e −1<n<1 is he s eng h o he ne wo k e ec (n=0 ep esen s he s anda d case
o non-ne wo k goods). Posi i e ( esp. nega i e) alues o n e lec a posi i e ( esp. nega i e)
consump ion ex e nali y. Though se e al ne wo k indus ies exhibi posi i e ex e nali ies
(mobile communica ions, so wa e, in e ne - ela ed ac i i ies, online social ne wo ks, ash-
ion, e c.), as a g ea e numbe o use s con ibu e o inc ease he alue o he p oduc o each
consume ( he e exis s a posi i e eedback loop i he ne wo k becomes mo e aluable), one
may hink abou eal-wo ld indus ies ha exhibi bo h kinds o ex e nali ies. In his ega d,
he au omobile indus y is wo h o be men ioned. This indus y migh also show nega i e
consump ion ex e nali ies as he mo e ca s a e sold, he g ea e he a ic conges ion, pa king
di icul ies and o he ela ed issues a e. The e o e, a nega i e ne wo k ex e nali y implies ha
an inc easing numbe o use s educes he alue o he goods o each consume ( o ins ance,
a ic conges ion o ne wo k conges ion o e limi ed bandwid h). The e m yi+yj ep esen s
he expec ed e ec i e ne wo k size o i m i’s consume s and o i m j’s consume s.
The solu ion o he maximisa ion p oblem o he su plus by he ep esen a i e consume
ollowing Ami e al. (2017) and Choné & Linneme (2020) gi es he ollowing linea in e se
demand o p oduc o ne wo k i:
pNC/NC
i=1−qi−qj+n(yi+yj)(A2)
8The ne wo k ex e nali y e ec in he u ili y unc ion Eq. (A.1) can mo e gene ally be w i en as n[qi(yi+
kiyj)+qj(yj+kjyi)]−n
2[y2
i+y2
j+(ki+kj)yiyj], whe e he pa ame e ki∈[0,1]measu es he deg ee o
compa ibili y o he ne wo k o p oduc j owa ds he ne wo k o p oduc i. Pai wise, he pa ame e kj∈[0,1]
measu es he deg ee o compa ibili y o he ne wo k o p oduc iwi h he ne wo k o p oduc j. Conside ing he
case o common compa ibili y ki=kj=k, he ne wo k ex e nali y e m becomes n[qi(yi+kyj)+qj(yj+
kyi)]−n
2(y2
i+y2
j+2kyiyj), which can be u he simpli ied by assuming ull compa ibili y be ween he
p oduc s o he wo ne wo ks, i.e., ki=kj=1, so ha n[qi(yi+yj)+qj(yj+yi)]−n
2(y2
i+y2
j+2yiyj).
I he deg ee o compa ibili y be ween he p oduc s o he wo ne wo ks was ze o, i.e., ki=kj=0, we would
ha e n(qiyi+qjyi)−n
2(y2
i+y2
j).
123
Annals o Ope a ions Resea ch
The mic oeconomic ounda ions o he ma ke demand
when bo h i ms commi hemsel es o an announced ou pu le el
This sec ion conside s he opposi e si ua ion in which bo h i ms c edibly commi hemsel es
oan announced ou pu le elbe o e consume s make hei pu chase decision. This ollows he
mechanism de ailed by Ka z & Shapi o (1985) in an ad hoc appendix o hei con ibu ion.
In his si ua ion, consume s know he quan i y ha will be p oduced by ha bo h i ms
( he consume s’ expec a ions abou he p oduc ion o he wo commi ing i ms a e ealised
immedia ely, i.e., a he same logical ime as he announcemen wi hou he need o know
he p e ailing equilib ium). As consume s conside he commi ed ou pu o bo h i ms,
subs i u ing ou yi=qiand yj=qjin o Eq. (A.2) gi es he linea in e se demand o
p oduc o ne wo k iunde C/C, ha is:
pC/C
i=1−qi−qj+n(qi+qj)(A3)
The mic oeconomic ounda ions o he ma ke demand when one i m
c edibly commi s i sel o an announced ou pu le el and he i al does
no
In he asymme ic scena io in which only i m, say i m i, can c edibly commi i sel o an
announced ou pu le el, consume s know he quan i y ha will be p oduced by ha i m ( he
consume s’ expec a ions abou he p oduc ion o he commi ing i m a e ealised imme-
dia ely, i.e., a he same logical ime as he announcemen wi hou he need o know he
p e ailing equilib ium), whe eas hey ha e expec a ions abou he ne wo k size o he non-
commi ing i m,say i m j.In hiscase,consume s’expec a ionsabou hequan i yp oduced
by i m jbecome binding, and hey a e ealised in equilib ium. Consume s a e a ional so
ha hey can know in ad ance (in he logical iming o he e en s) he p e ailing equilib ium
alue o he ou pu . Knowing ha consume s conside he commi ed ou pu only o i m i,
subs i u ing ou yi=qiin o Eq. (A.2) gi es he linea in e se demand o p oduc o ne wo k
iunde C/NC, ha is:
pC/NC
i=1−qi−qj+n(qi+yj)(A4)
P oo o P oposi ion 1
Le nbe posi i e. I 0 ≤b<0.219 hen A(n,b)>0, B(n,b)<0andC(n,b)>
0 o any0≤n<nC(b);A(n,b)>0, B(n,b)<0andC(n,b)<0 o
any nC(b)<n<1 so ha Poin [1] holds. I 0.219 <b<0.313 hen A(n,b)<0,
B(n,b)>0andC(n,b)>0 o any0≤n<nB(b);A(n,b)<0,B(n,b)<
0andC(n,b)>0 o anynB(b)<n<nA(b);A(n,b)>0, B(n,b)<0and
C(n,b)>0 o anynA(b)<n<nC(b);A(n,b)>0, B(n,b)<0and
C(n,b)<0 o anynC(b)<n<1 so ha Poin [2] holds. I 0.313 <b<0.393
hen A(n,b)<0, B(n,b)>0andC(n,b)>0 o any0≤n<nA(b);
A(n,b)>0, B(n,b)>0andC(n,b)>0 o anynA(b)<n<nB(b);
A(n,b)>0, B(n,b)<0andC(n,b)>0 o anynB(b)<n<nC(b);
A(n,b)>0, B(n,b)<0andC(n,b)<0 o anynC(b)<n<1 so ha Poin
123
Annals o Ope a ions Resea ch
[3] holds. I 0.393 <b<0.4013 hen A(n,b)<0, B(n,b)>0andC(n,b)>
0 o any0≤n<nA(b);A(n,b)>0, B(n,b)>0andC(n,b)>0o
C(n,b)<0 o anynA(b)<n<nB(b);A(n,b)>0, B(n,b)<0and
C(n,b)<0 o anynB(b)<n<1 so ha Poin [4] holds. I 0.4013 <b<0.5 hen
A(n,b)<0,B(n,b)>0andC(n,b)>0 o any0≤n<nA(b);A(n,b)>
0, B(n,b)>0andC(n,b)>0o C(n,b)<0 o anynA(b)<n<1so ha
Poin [5] holds. Le nbe nega i e. I 0 ≤b<0.115 hen A(n,b)<0, B(n,b)>0
and C(n,b)<0 o any0>n>−1. I 0 ≤b<0.115 hen A(n,b)<0,
B(n,b)>0andC(n,b)<0 o any0>n>−1 so ha Poin [6] holds. I
0.115 <b<0.168 hen A(n,b)<0, B(n,b)>0andC(n,b)<0 o any
0>n>−nB(b);A(n,b)<0, B(n,b)<0andC(n,b)<0 o any−nB(b)>
n>−1 so ha Poin [7] holds. I 0.168 <b<0.219 hen A(n,b)<0, B(n,b)>0
and C(n,b)<0 o any0>n>−nB(b);A(n,b)<0, B(n,b)<0and
C(n,b)<0 o any−nB(b)>n>−nA(b);A(n,b)>0, B(n,b)<0and
C(n,b)<0 o any−nA(b)>n>−1 so ha Poin [8] holds. I 0.219 <b<0.333
hen A(n,b)>0, B(n,b)<0andC(n,b)<0 o any0>n>−1so ha
Poin [9] holds. I 0.333 <b<bNC/NC
T(n) hen A(n,b)>0, B(n,b)<0and
C(n,b)<0 o any0>n>−1−2b
bso ha Poin [10] holds. Q.E.D.
The CSR-CDG in a S ackelbe g en i onmen
The CSR-CDG is a mul i-s age non-coope a i e game wi h comple e in o ma ion in which,
in he i s decision-making s age, each i m chooses o c edibly commi o no o commi
o an announced ou pu le el. This is a bina y choice ha happens be o e consume s make
hei pu chase decisions and hen a ec s hei expec a ions abou he p e ailing ne wo k
size. Consume s a e a ional and ha e a ional expec a ions. I a leas one o he wo i ms is
commi ing, he consume s’ expec a ions abou he p oduc ion o he commi ing i m(s) a e
ealised immedia ely, i.e., a he same logical ime as he announcemen wi hou he need o
know he p e ailing equilib ium. I a leas one o he wo i ms is no commi ing, consume s’
expec a ionsbecomebinding,and heya e ealisedinequilib iumasconsume sa e a ionalso
ha hey can know in ad ance (in he logical iming o he e en s) he p e ailing equilib ium
alue o he ou pu . This hold in an in e media e s age be ween he decision-making s age and
he ma ke s age. In he ma ke s age, compe i ion be ween i ms can occu simul aneously
(Cou no ) o sequen ially (S ackelbe g). In he o me case, any decision o each i m can
only be expec ed by he i al because he logical iming o e en s is such ha no playe can
obse e each o he ’s decisions ega dless o whe he a i m c edibly commi s i sel o an
announced ou pu le el. This case has been analysed in he main ex o he a icle. In he
la e case, one i m ( he leade ) chooses be o e i s i al, who is he e o e o ced o become
a ollowe choosing i s p oduc ion a e ha ing obse ed he p oduc ion – and he decision
o c edibly commi o no o commi – by he leade . This sec ion deals wi h his case and
de elops he CSR-CDG in which a he ma ke s age quan i y compe i ion occu s sequen ially
acco ding o he S ackelbe g ules.
Le us assume ha i m iis he leade and i m jis he ollowe . The game is sol ed by
using he backwa d induc ion logic and conside s p o i s as he main a iable a he decision-
making s age. The s uc u e o he ou sub-games is he ollowing: (i) bo h i ms c edibly
commi hemsel es o an announced ou pu le el (C/C), he leade commi s i sel be o e
he ollowe does; (ii) bo h i ms do no (o canno ) commi hemsel es o an announced
123
Annals o Ope a ions Resea ch
s a egy Co s a egy NC. In addi ion, we ha e also assumed, in he sub-game NC/Co he
S ackelbe g model p esen ed in he p e ious sec ion o his appendix, in which he leade is
no commi ing, ha , he c edible public announcemen o ou pu commi men by he by he
C- i m occu s ex-pos , i.e., a e he i al selec ed s a egy NC.
The e o e, acco ding o he e iewe ’s poin , he symme ic sub-games can be played
acco ding o he Cou no ules bu in he asymme ic sub-games he C- i m (ei he i m i
o i m j) always becomes a S ackelbe g leade . This implies ha he equilib ium esul s o
he asymme ic sub-game C/NC ob ained by conside ing a S ackelbe g scena io in which
i m iis he leade a e he same as hose eme ging in he model o he p esen sec ion. This
implies ha CSR-CDG à la S ackelbe g is an asymme ic game and he CSR-CDG à la
Cou no -S ackelbe g is a symme ic game.
To sum up, in he Cou no -S ackelbe g CSR-CDG bo h C- i ms in he symme ic sub-
game C/Cpublic announce, o some eason, hei ou pu commi men in he same logical
iming o he e en s, whe eas he C- i m in he asymme ic sub-games C/NC and NC/C
public announces, o some eason, i s ou pu commi men be o e he i al’s akes any ac ions
and hen he C- i m becomes a S ackelbe g leade . O cou se, his modi ies he logical iming
o he e en s occu ing (simul aneously) in he symme ic sub-games om hose occu ing
(sequen ially) in he asymme ic sub-games.
The payo ma ix (p o i s) ha i ms ace in he i s decision-making s age o he CSR-
CDG played in a Cou no -S ackelbe g en i onmen is composed o Eqs. (13), (20), (A.15)
and (A.16). The SPNE o he game a e ob ained by compu ing he usual p o i di e en ials
ha we do no epo he e o sa e space. The main esul s a e summa ised in Fig. A2 , which
igo ously epo s he lociin which hep o i di e en ialsa eze oand he easibili ycondi ion
ha mus besa is ied ogua an eeposi i ep o i sin hesub-game NC/NC.The igu e epo s
he loci o poin s such ha CS
A(n,b)=0, CS
B(n,b)=0andCS
C(n,b)=0, i.e.,
nCS
A(b),nCS
B(b)and nCS
C(b) espec i ely, whe e he supe -sc ip CS s ands o “Cou no -
S ackelbe g”. Resul s a e quali a i ely like hose ob ained in he Cou no model o he main
ex and he S ackelbe g model o he p e ious sec ion o he appendix.
Posi i e ne wo k e ec s
•A ea A:(C,C)is he unique SPNE (CS
A(n,b)>0, CS
B(n,b)<0and
CS
C(n,b)>0), which is Pa e o ine icien . The CSR-CDG à la Cou no -S ackelbe g
is a p isone ’s dilemma wi h commi ing i ms.
•A ea B:(C,C)is he unique SPNE (CS
A(n,b)>0, CS
B(n,b)<0and
CS
C(n,b)<0), which is Pa e o e icien . The CSR-CDG à la Cou no -S ackelbe g
is an an i-p isone ’s dilemma wi h commi ing i ms.
•A ea C:(C,C)and (NC,NC)a e wo symme ic SPNE (CS
A(n,b)<0,
CS
B(n,b)<0andCS
C(n,b)>0), bu NC payo domina es C. The CSR-CDG à
la Cou no -S ackelbe g is a coo dina ion game.
•A ea D:(NC,NC)is he unique SPNE (CS
A(n,b)<0, CS
B(n,b)>0and
CS
C(n,b)>0), which is Pa e o e icien . The CSR-CDG à la Cou no -S ackelbe g
is an an i-p isone ’s dilemma wi h non-commi ing i ms.
•A ea Eand A ea F:(C,NC)and (NC,C)a e wo asymme ic SPNE (CS
A(n,b)>0,
CS
B(n,b)>0), which a e Pa e o e icien , and he C- i m ge s he bes ou come.
Nega i e ne wo k e ec s
•A ea G:(C,C)is he unique SPNE (CS
A(n,b)>0, CS
B(n,b)<0and
CS
C(n,b)<0), which is Pa e o e icien . The CSR-CDG à la Cou no -S ackelbe g
is an an i-p isone ’s dilemma wi h commi ing i ms.
123

Annals o Ope a ions Resea ch
Fig. A.2 The CSR-CDG à la Cou no -S ackelbe g: SPNE when nand b a y and he i m’s p o i is he decision
a iable.Thesand-colou ed egion ep esen heun easiblepa ame e space,whichisbounded by he h eshold
b<1
2−n:= bNC/NC
T(n). The easibili y condi ion mus be sa is ied o gua an ee posi i e p o i s in he sub-
game NC/NC. Ligh -blue egion (F): (C,NC)and (NC,C)a e wo asymme ic SPNE wi h he C- i m
ge ing he bes ou come and he CSR-CDG à la Cou no -S ackelbe g is an an i-coo dina ion game. This
acco ds wi h he egion below (E), in which he CSR-CDG à la Cou no -S ackelbe g is an an i-coo dina ion
game and C- i m con inues o ge he bes ou come
•A ea H:(C,C)and (NC,NC)a e wo symme ic SPNE (CS
A(n,b)<0,
CS
B(n,b)<0andCS
C(n,b)<0), bu Cpayo domina es NC. The CSR-CDG à
la Cou no -S ackelbe g is a coo dina ion game.
•A ea I:(C,C)and (NC,NC)a e wo symme ic SPNE (CS
A(n,b)<0,
CS
B(n,b)<0andCS
C(n,b)<0), bu Cpayo domina es NC. The CSR-CDG à
la Cou no -S ackelbe g is a coo dina ion game.
•A ea L:(C,C)is he unique SPNE (CS
A(n,b)>0, CS
B(n,b)<0and
CS
C(n,b)<0), which is Pa e o e icien . The CSR-CDG à la Cou no -S ackelbe g
is an an i-p isone ’s dilemma wi h commi ing i ms.
Cou no e sus S ackelbe g
This sec ion discusses some possible ad ancemen s o he modelling se up used in he p esen
pape by speci ically concen a ing on he compa ison be ween he simul aneous Cou no
se ing and he sequen ial S ackelbe g se ing. The discussion begins by conside ing he
delega ion li e a u e amed in s a egic compe i i e ma ke s de eloped by Fe sh man and
Judd (1987), Skli as (1987) and Vicke s (1985), FJVS hence o h.
The manage ial delega ion game à la FJSV ep esen s a classic example in which he
choices o he playe s a e always simul aneous in all sub-games, i.e., he scena io is he
Cou no -Nash one. This implies ha in he asymme ic sub-game whe e, o example, he
owne o i miis delega ing he ou pu o a manage and i m jis p o i maximizing, he
123
Annals o Ope a ions Resea ch
Fig. A.3 The CSR-CDG: Cou no e sus S ackelbe g. The ed line is he cu e de ining, in he space (b,n),
he indi e ence o play he CSR-CDG acco ding o he Cou no -Nash (simul aneous) ules o acco ding o
he S ackelbe g (sequen ial) ules wi h posi i e and nega i e alues o n
same payo s (p o i s) a e ob ained as in he case whe e he delega ing i m iis he leade
and he p o i -maximising i m jis he ollowe in he p oduc ma ke like in a S ackelbe g
scena io. This, in u n, implies ha p o i s unde Cou no -Nash o S ackelbe g coincide (see,
o example, he payo ma ix in Fan i e al., 2017,Table1, p. 498, whe e his esul is clea ly
illus a ed).
Howe e , in a game à la FJSV, he SPNE is always he one in which bo h i ms choose he
Cou no -Nash delega ion s a egy (so he asymme ic case in which one i m is delega ing
and he i al is p o i -maximising does no eme ge as a possible SPNE o his kind o endoge-
nous s a egic delega ion games). The e o e, he o e lapping o p o i s be ween Cou no and
S ackelbe g is no s a egically in e es ing in ha case. Indeed, i is also well known ha ade-
qua e changes o he FJSV model may imply, in he asymme ic sub-game in which one i m
is delega ing and he i al is p o i -maximising, ha i i ms choose acco ding o he S ackel-
be g ules (sequen ially) and no acco ding o he Cou no -Nash ule (simul aneously) hen
unde app op ia e pa ame ic condi ions he s a egic si ua ion o mixed choices (in which
i is assumed compe i ion à la S ackelbe g) may eme ge as he SPNE o he game: his is
he case in Basu’s (1995) con ibu ion. Fo he sake o p ecision, we no e ha Basu’s (1995)
model is e y speci ic in he s a egic delega ion li e a u e à la FJSV, as esul s equi e he
p esence o ixed hi ing manage ial cos s, which a e commonly neglec ed.
The analysis below aims o cla i y he ollowing ques ion. Is i possible ha he quan i ies
chosen simul aneously in he p oduc ma ke gene a e he o e lapping o he co esponding
Cou no -Nash payo s (p o i s) wi h he payo s (p o i s) ob ained in a sequen ial S ackelbe g
scena io in which he commi ing C- i m is he leade and he non-commi ing NC- i m is
he ollowe ? The answe is posi i e, bu his applies o a speci ic pa ame ic se . Indeed,
he e exis s a cu e de ined by he equa ion x(b,n)=0 such ha he p o i s ollowing he
simul aneous Cou no -Nash scena io in which i m iis playing Cand i m jis playing NC
coincide wi h he p o i s ollowing he sequen ial S ackelbe g scena io in which he C- i m i
123
Annals o Ope a ions Resea ch
is he leade and he NC- i m jis he ollowe . This unc ion is compu ed as he di e ence
be ween he p o i s o i m i( esp. i m j) in he asymme ic sub-game o he Cou no -Nash
scena io, as gi en by Eq. (33) ( esp. Equa ion (34)) in he main ex and he p o i s o i m i
( esp. i m j) in he asymme ic sub-game o he S ackelbe g scena io in which he C- i m
is he leade and he NC- i m is he ollowe , as gi en by Eq. (A.15) ( esp. Eq. (A.16)).
The cu e x(b,n)=0, as depic ed in Fig. A3 (in which he solu ions in he space (b,n)o
he p o i di e en ials o i m iand i m jdiscussed so a jus o e lap), is de ined only in
an implici o m as i is no possible o explici nas a unc ion o bo ice e sa in a nea
analy ical o m. Thus, only o he pa ame ic se in he space (b,n)de ined by x(b,n)does
he Cou no -Nash simul aneous choice hold as i i we e a S ackelbe g sequen ial se -up a
he ma ke s age. This esul o ou commi men decision game exac ly esembles ha o
Basu (1995) o his manage ial delega ion game.
This also means ha a pa ame ic egion ( o ins ance, abo e he cu e de ined by he ed
line x(b,n)=0 in Fig. A3 o posi i e alues o n) does exis in which he S ackelbe g ou -
come a ises endogenously h ough he commi men s a egy wi hin he commi men decision
game like he delega ion s a egy eme ges in he s a egic delega ion game o Basu (1995).
This basic analysis aises ques ions ha can be o impo ance abou he mode o com-
pe i ion (endogenous o de o mo es) ha can eme ge as a Nash ou come (SPNE) o an
endogenous iming game à la Hamil on and Slu sky (1990) wi h CSR commi ing and non-
commi ing i ms in a ne wo k indus y. I he e en s happen simul aneously, he compe i ion
in he sub-game holds à la Cou no . I he e en s happen sequen ially, he compe i ion in
he sub-game holds à la S ackelbe g. The endogenous SPNE o he game can be ei he he
simul aneous mo e o he sequen ial mo e. This imp o emen is le o u u e esea ch.
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